Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3973 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,9)(4,7)(5,8)(6,12)(10,14)(11,15)(13,16)(17,18)(19,24)(20,27)(21,25)(22,23)(26,31)(28,32)(29,30)(34,38)(35,43)(36,46)(42,48);(1,3,7,5)(2,8,4,9)(6,13,14,11)(10,16,12,15)(18,23)(20,24)(21,29)(26,32)(33,34,39,38)(35,44,43,40)(36,41,46,47)(37,48,45,42);(2,4)(8,9)(10,12)(15,16)(17,19,22,27)(18,24,23,20)(21,26,29,32)(25,31,30,28)(33,35,39,43)(34,40,38,44)(36,45,46,37)(41,48,47,42);(1,4,7,2)(3,9,5,8)(6,12,14,10)(11,16,13,15)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,32)(30,31)(33,36)(34,41)(35,45)(37,43)(38,47)(39,46)(40,48)(42,44);(1,5,7,3)(2,9,4,8)(6,13,14,11)(10,16,12,15)(17,20)(18,19)(21,28)(22,24)(23,27)(25,26)(29,31)(30,32)(33,37,39,45)(34,42,38,48)(35,36,43,46)(40,41,44,47);(1,6,7,14)(2,10,4,12)(3,11,5,13)(8,15,9,16)(17,21,22,29)(18,25,23,30)(19,26,27,32)(20,28,24,31)(33,38,39,34)(35,44,43,40)(36,47,46,41)(37,48,45,42)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662593);